In a documentation, there is the following formula for "zero interest rate risky duration" of a bond: $\frac{1-exp(-s \cdot T)}{s}$, where $s$ is spread, $T$ time until maturity.

What type of bond (zero-coupon, floating rate, etc.) is this formula valid for and how is it derived?


1 Answer 1


Here is my understanding of your question, I might have oversimplified your problem, and made some hypothesis that were not yours.

Assuming we are talking about a bond paying $1$ each day until $T$ or default event if occuring before $T$.

Let's write the risky coupon bond payment in a continous time manner:

$$\int_0^T \mathbb{1}_{\tau>t} dt$$

with zero interest rate, NPV is given by (assuming flat intensity)

$$NPV = \int_{0}^{T} \mathbb{E}[\mathbb{1}_{\tau>t}]dt = \int_{0}^{T} e^{-st}dt=\frac{1-e^{-sT}}{s}$$

We focus on risky duration as derivative of the NPV with respect to spread.

Here, we are in a continuous time framework where we assumed no recovery in NPV, so intensity and spread are the same

$$\text{risky duration}=\frac{dNPV}{ds}=\frac{(1+sT)e^{-sT}-1}{s^2}=-\frac{T^2}{2}+O_{s\to 0}(sT^3)]$$

  • $\begingroup$ Thanks, but what I meant is a bond whose risky duration is equal to $\frac{1−exp(−sT)}{s}$. $\endgroup$
    – Dello
    Jun 9, 2016 at 18:06

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.